Let A_n be the axiom "all polynomials of degree n have roots", for an
arbitrary field K. While generalizing the Fundamental Theorem of
Algebra, I had established various finite implications between these
axioms, mainly following from (A_k & A_d) --> A_n whenever d = (n
choose k) = n!/(k!(n-k!)).
(In characteristic p you may additionally need to assume A_p).
But I thought there might be deeper examples, based on the kind of
combinatorics of permutation groups that John Conway explored in his
work on finite versions of the Axiom of Choice in 1970. I went to see
Conway today, and we were able to establish, for example, A30 --> A8,
and (A3 & A10) --> A6, which go beyond what my binomial coefficient
techniques could show.
A related phenomenon: if A6 is true (all sextic equations have roots),
then not only can you get A2 (trivially), A3 (trivially), and A4
(because quartics can be reduced to quadratics and cubics), but any
quintic which does NOT have roots must have a cyclic Galois group
(because of combinatorics involving transitive representations of S5 in
S6).
It looks like there is a fruitful area for research here. It's curious
no one seems to have examined these finitary relationships!
-- JS